3.6 Summary
Key Points
- To represent numbers we need a fixed, ordered standard sequence of objects, names, or symbols, a number system, and we count by building a one-to-one mapping with it.
- The written symbols of a number system are called numerals.
- Landmark numbers are special reference numbers (like Roman I, V, X, L, C, D, M) used to group and represent other numbers, especially large ones.
- A system whose landmark numbers are the powers of a number n is a base-n system; its landmark numbers are 1, n, n^2, n^3, \dots Base 10 is the decimal system.
- A base system that uses the position of each symbol to fix which landmark number it counts is a positional or place value system. Place value was reached by the Mesopotamian, Mayan, Chinese, and Indian peoples.
- The number system used worldwide today is the Hindu (Indian / Hindu–Arabic) number system, a base-10 place value system with ten digits including 0. Treating 0 as a number lets every number be written unambiguously with finitely many symbols and makes computation efficient. It arose in India about 2000 years ago and ranks among humanity’s greatest inventions.